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104       XIII    INTEGRATION

A measure /<( on X is said to be real if p = //, or equivalently if jLi(f) is real
for every/e Jf R(X).

We may therefore identify the set of real measures on X with a vector
space of linear forms on the real vector space JTR(X). This vector space is
denoted by MR(X). Lebesgue measure and all Dirac measures are real. If /JL
is any complex measure, then the measures /^ = (/* + /I)/2 and ju2 = GU  fi)/2i
are real. They are called respectively the real and imaginary parts of ju, and
are denoted by 0t\i and J\i respectively. For each function /e JTR(X), we
have

(13.2.1)

and by definition

(13.2.2)

3. POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE

A measure n on a locally compact space X is said to be positive if, for
each function fe ^TR(X) such that /^ 0, we have //(/) ^ 0. Consequently,
if /and g are two functions belonging to JTR(X) such that/^^, we have
X/) ^ /((#). Since each/e #*R(X) can be written in the form /=/+ -/"
(where f+(x) = (/(x))+ and /" = (/(x)) (2.2)), it follows that a posi-
tive measure is a raz/ measure. We denote by M + (X) the set of all positive
measures on X.

Surprisingly, the property of positivity alone implies the defining property
(13.1.2) of a measure:

(13.3.1)   Let n be a linear form on the real vector space 3fR(X) such that
//(/) *> 0 whenever /^ 0. Then /i is a (positive) measure on X.

We have to show that (1 3.1 .2) is satisfied. There exists a function g e ^R(X)
with values in [0, 1] and equal to 1 throughout K ((3.18.2) and (4.5.2)).
Hence for all/e <?TR(X; K), we have

and therefore

o ^ jjL(f+) z \\f\\  fa),     o g x/-) ^ ll/ll

so that finally |X/)I^ 2 ||/|| by /L We have Ji = ^, and if A, ^ are measures on X and a, b are